Apparatus for cooling of electronic devices utilizing microfluidic components

ABSTRACT

System and method for cooling of an integrated circuit utilizing Micro-Electro Mechanical Systems (MEMS) components. A flexible thin film has an upper flexible substrate, a lower inflexible substrate and flexible seals. One face of the lower substrate is in contact with at least one hot medium and the other face is in contact with a coolant fluid. One face of the upper substrate is in contact with the coolant fluid and the other face is in contact with the surrounding ambient. Two continuous flexible seals are attached to the faces of the upper lower substrates to form at least one closed enclosure comprising a thermally conducting gas. The thermally conducting gas is in direct contact with the lower substrate. The upper substrate deflects continuously and maximally in the direction along the coolant fluid flow direction when the flexible seals deflect when the thermally conducting gas undergoes volumetric thermal expansion.

FIELD OF THE INVENTION

Embodiments are generally related to thin film channels. Embodimentsalso relate to the field of microfluidic devices and cooling ofelectronic devices. Embodiments additionally relate to system and methodfor cooling of an integrated circuit utilizing Micro-Electro MechanicalSystems (MEMS) components.

BACKGROUND OF THE INVENTION

Various engineering applications utilize fluidic thin films as coolingdevices. Examples include heat pipes and microchannel heat sinks.

Such devices are widely utilized in electronic cooling applicationswhere rapid developments in the electronics industry necessitates acontinuing need for increasing cooling capacity. Several solutions havebeen proposed to increase the cooling capacity of fluidic thin films.For example, some solutions have demonstrated that two-phase flow in aminichannel is capable of removing maximum heat fluxes generated byelectronic packages however the system may become unstable near certainoperating conditions. Further, it has been found that the use of porousmedium in cooling of electronic devices can enhance heat transfer.However, a porous medium can create a substantial increase in thepressure drop inside the thin film.

Recently, flexible fluidic thin films have been introduced for enhancingthe cooling capability of fluidic thin films. Flexible thin filmsutilize soft seals to separate between their plates instead of havingrigid thin film construction. It has also been demonstrated that morecooling is achievable when flexible fluidic thin films are utilized. Theexpansion of the flexible thin film including flexible microchannel heatsink is directly related to the average internal pressure inside themicrochannel. Additional increases in the pressure drop across aflexible microchannel not only increases the average velocity, but alsoexpands the microchannel causing an apparent increase in the coolantflow rate which, in turn, increases the cooling capacity of the thinfilm.

It has also been demonstrated that the cooling effect of flexible thinfilms can be further enhanced if the supporting soft seals containclosed cavities filled with a gas, which is in contact with the heatedsubstrate boundary of the thin film. Such special kind of sealingassembly is referred as “flexible complex seals”. The resulting fluidicthin film device is expandable according to an increase in the workinginternal pressure or an increase in the heated substrate temperature.

Flexible thin films have been analyzed and considered in the context ofspecial designs that result in a uniform displacement of the thin filmmobile substrate. In contrast, the displacement of a mobile substrate inother flexible thin film designs can be non-uniform especially if themobile substrate itself is flexible.

Therefore, it is believed that a need exists for improved thermallyexpandable flexible fluidic thin films that can be utilized inMicro-Electro Mechanical Systems (MEMS) and electronic coolingapplications such as those involving integrated circuits, servers, andso forth. Also such films should produce significant increase in coolingas the heating load increases especially when operated at lower Pecletnumbers.

SUMMARY OF THE INVENTION

The following summary is provided to facilitate an understanding of someof the innovative features unique to the disclosed embodiment and is notintended to be a full description. A full appreciation of the variousaspects of the embodiments disclosed herein can be gained by taking theentire specification, claims, drawings, and abstract as a whole.

It is, therefore, one aspect of the disclosed embodiments to provide forthin film channels in the context of cooling electronic devices, suchas, for example, integrated circuits, computers, servers, and the like.

It is another aspect of the disclosed embodiments to provide formicrofluidic devices and the cooling of electronic devices.

It is yet another aspect of the disclosed embodiments to provide for asystem and method for cooling electronic devices utilizing Micro-ElectroMechanical Systems (MEMS) components.

The aforementioned aspects and other objectives and advantages can nowbe achieved as described herein. A flexible thin film includes an upperflexible substrate, a lower inflexible substrate and flexible seals. Oneface of the lower substrate is in contact with at least one hot mediumand the other face is in contact with a coolant fluid. The “hot medium”can be, for example, an electronic device. One face of the uppersubstrate is in contact with the coolant fluid and the other face is incontact with the surrounding ambient. Two continuous flexible seals areattached to the faces of the upper substrate and the lower substrate toform at least one closed enclosure comprising a thermally conductinggas. The thermally conducting gas is in direct contact with the lowersubstrate. The upper substrate deflects continuously and maximally inthe direction along the coolant fluid flow direction when the flexibleseals deflect and the thermally conducting gas undergoes volumetricthermal expansion.

Heat transfer inside thermally expandable fluidic and flexible thinfilms is analyzed. The upper flexible substrate of the thin film isallowed to expand as functions of both the pressure inside the thin filmand the local heated substrate temperature via utilizing flexiblecomplex seals. The expansion of the thin film is considered to be in thetransverse direction of the fluid flow direction. The expansion in thethin film heights is generally related to both the local internalpressure and the local heated substrate temperature. The governingequations for flow and heat transfer can be properly non-dimensionalizedand reduced into simpler forms. The resulting equations can be thencomputationally solved and various pertinent results obtained. Thecontrolling parameters can be obtained and their role regarding thethermal behavior of thermally expandable fluidic flexible thin films canbe established.

DESCRIPTION OF THE DRAWINGS

The accompanying figures, in which like reference numerals refer toidentical or functionally-similar elements throughout the separate viewsand which are incorporated in and form a part of the specification,further illustrate the disclosed embodiments and, together with thedetailed description of the invention, serve to explain the principlesof the disclosed embodiments.

FIG. 1A illustrates a side view of a thermally expandable flexible thinfilm with mobile flexible upper substrate, in accordance with thedisclosed embodiments;

FIG. 1B illustrates a sectional view of a thermally expandable flexiblethin film with mobile flexible upper substrate of FIG. 1A, in accordancewith the disclosed embodiments;

FIG. 2 illustrates a graph depicting the variation of the dimensionlessthin film height H with the thermal expansion parameter FT1, inaccordance with the disclosed embodiments;

FIG. 3 illustrates a graph depicting the variation of the dimensionlessmean bulk temperature θ_(m) and dimensionless lower substratetemperature θ_(B) with the thermal expansion parameter F_(T1), inaccordance with the disclosed embodiments;

FIG. 4 illustrates a graph depicting the variation of the dimensionlessthin film height H with Peε, in accordance with the disclosedembodiments;

FIG. 5 illustrates a graph depicting the variation of the dimensionlessmean bulk temperature θ_(m) and dimensionless lower substratetemperature θ_(B) with Peε, in accordance with the disclosedembodiments;

FIG. 6 illustrates a graph depicting the variation of the lowersubstrate temperature at the exit θ_(B)(F_(T1),X=1) relative to thatwhen F_(T1)=0 with Peε and F_(T), in accordance with the disclosedembodiments; and

FIG. 7 illustrates a graph depicting the, variation of the averageNusselt number Nu_(ATC)(F_(T1),X=1) relative to that when F_(T1)=0 withPeε and F_(T1), in accordance with the disclosed embodiments.

DETAILED DESCRIPTION OF THE INVENTION

The particular values and configurations discussed in these non-limitingexamples can be varied and are cited merely to illustrate at least oneembodiment and are not intended to limit the scope thereof.

The following Table 1 provides various symbols and meanings, which canbe utilized in the context of the disclosed embodiments:

TABLE 1 Nomenclature B thin film length [m] D half thin film width [m]D_(o) reference half thin film width [m] c_(p) specific heat of thefluid [J/kg K] F_(T1) thermal expansion parameter for flexible complexseal F_(T2) thermal expansion parameter for side biomaterial sidesupports H dimensionless thin film height H₁ dimensionless inlet thinfilm height h thin film height [m] h_(o) reference thin film height [m]h_(c) convective heat transfer coefficient [W/m² K] k thermalconductivity of the fluid [W/m K] Nu Nusselt number P pressure [Pa]P_(i) inlet pressure [Pa] P_(e) exit pressure [Pa] Pe Peclet number Pfluid pressure [Pa] q heat flux at the lower substrate [W/m²] S_(1,2)stiffness parameter for flexible complex seal or bimaterial plate T,T_(i) temperature in fluid and the inlet temperature [K] U, udimensionless and dimensional axial velocities u_(o) reference axialvelocity [m/s] V, v dimensionless and dimensional normal velocities W, wdimensionless and dimensional spanwise velocities X, x dimensionless anddimensional axial coordinates Y, y dimensionless and dimensional normalcoordinates Z, z dimensionless and dimensional spanwise coordinatesGreek Symbols ε aspect ratio μ dynamic viscosity of the fluid θ, θ_(m)dimensionless temperature and dimensionless mean bulk temperature θ_(B)dimensionless temperature at the lower substrate ρ density of the fluid[kg/m³] η variable transformation for the dimensionless Y-coordinate

A two dimensional thin film having a small height h compared to itslength B can be considered. The x-axis can be taken in the direction ofthe length of the thin film while y-axis is taken along the height ofthe thin film as shown in FIGS. 1A and 1B. The z-axis can be taken alongthe half thin film width, D, where it is large enough such thattwo-dimensional flow within the thin film can be assumed. The thin filmheight and width can be allowed to vary with the axial distance. Theflow is assumed to be laminar and the convective terms are neglectedcompared to diffusion terms in the momentum equation (hydrodynamicallyfully developed flow is assumed). Under these conditions, the velocityfield, Reynolds equation and the energy equation can be obtained asfollows:

$\begin{matrix}{{X = \frac{x}{B}};{Y = \frac{y}{h_{o}}};{Z = \frac{z}{D_{o}}}} & {{{Eqs}.\mspace{14mu} 6}\left( {a,b,c} \right)} \\{{U = \frac{u}{u_{o}}};{V = \frac{v}{u_{o}\left( {h_{o}/B} \right)}};{W = \frac{w}{u_{o}\left( {D_{o}/B} \right)}}} & {{{Eqs}.\mspace{14mu} 6}\left( {d,e,f} \right)} \\{{\theta = \frac{T - T_{1}}{{qh}_{o}/k}};{\overset{\_}{P} = \frac{P - P_{e}}{P_{i} - P_{e}}}} & {{{Eqs}.\mspace{14mu} 6}\left( {g,h} \right)} \\{{H = \frac{h}{h_{o}}};{\overset{\_}{D} = \frac{D}{D_{o}}}} & {{{Eqs}.\mspace{14mu} 6}\left( {i,k} \right)}\end{matrix}$

where the variables T, u, v, w, ρ, P, μ, c_(p) and k represent,respectively, the fluid temperature, axial velocity, normal velocity,velocity component in the z-direction, fluid's density, pressure,fluid's dynamic viscosity, fluid's specific heat and the fluid's thermalconductivity.

The dimensionless variables are as follows:

$\begin{matrix}{{X = \frac{x}{B}};{Y = \frac{y}{h_{o}}};{Z = \frac{z}{D_{o}}}} & {{{Eqs}.\mspace{11mu} 6}\left( {a,b,c} \right)} \\{{U = \frac{u}{u_{o}}};{V = \frac{v}{u_{o}\left( {h_{o}\text{/}B} \right)}};{W = \frac{w}{u_{o}\left( {D_{o}\text{/}B} \right)}}} & {{{Eqs}.\mspace{11mu} 6}\left( {d,e,f} \right)} \\{{\theta = \frac{T - T_{1}}{q\; h_{o}\text{/}k}};{\overset{\_}{P} = \frac{P - P_{e}}{P_{i} - P_{e}}}} & {{{Eqs}.\mspace{11mu} 6}\left( {g,h} \right)} \\{{H = \frac{h}{h_{o}}};{\overset{\_}{D} = \frac{D}{D_{o}}}} & {{{Eqs}.\mspace{11mu} 6}\left( {i,k} \right)}\end{matrix}$

where q, u_(o), T₁, h_(o) and D_(o) are the heat flux at the lowersubstrate, reference velocity, inlet temperature, a reference thin filmheight and a reference half thin film width, respectively. The referencethin film height h_(o) and width D_(o) are taken to be the thin filmheight and width at the inlet when both the flow and the heat flux inthe thin film are zero. The Reynolds and energy equations can be writtenas

$\begin{matrix}{{{\left( \frac{D_{o}}{B} \right)^{2}\frac{\partial}{\partial X}\left( {\left\lbrack {H(X)} \right\rbrack^{3}\frac{\partial\overset{\_}{P}}{\partial X}} \right)} + {\frac{\partial}{\partial Z}\left( {\left\lbrack {H(X)} \right\rbrack^{3}\frac{\partial\overset{\_}{P}}{\partial Z}} \right)}} = 0} & {{Eq}.\mspace{11mu} 7} \\{{{U\frac{\partial\theta}{\partial X}} + {V\frac{\partial\theta}{\partial Y}} + {W\frac{\partial\theta}{\partial Z}}} = {\frac{1}{{Pe}\; ɛ}\frac{\partial^{2}\theta}{\partial Y^{2}}}} & {{Eq}.\mspace{11mu} 8}\end{matrix}$

where the Peclet number (Pe) and the aspect ratio (ε) are defined as

$\begin{matrix}{{{Pe} = \frac{\rho \; c_{p}u_{o}h_{o}}{k}};{ɛ = \frac{h_{o}}{B}}} & {{{Eqs}.\mspace{11mu} 9}\left( {a,b} \right)}\end{matrix}$

Referring to FIGS. 1A and 1B, a flexible thin film 100 includes an uppersubstrate 102 and a lower substrate 104. The lower substrate 104comprises an inflexible substrate and the upper substrate 102 comprisesa flexible substrate. The one face of the lower substrate 104 isgenerally in contact with at least one “hot medium” and the other faceis in contact with a coolant fluid 110. The one face of the uppersubstrate 102 is in contact with the coolant fluid 110 and the otherface is in contact with the surrounding ambient. The faces of the upperand lower substrates 102 and 104 are in contact with the coolant fluid110 and oppose each other. Two flexible seals 106 and 108 can beattached to the faces of the upper substrate 102 and the lower substrate104 opposing the coolant fluid 110 to form one or more closed enclosurescontaining a thermally conducting gas.

FIG. 1B is a sectional view of a thermally expandable flexible thin filmwith mobile flexible upper substrate 102 taken along A-A of FIG. 1A, inaccordance with the disclosed embodiments. The thermally conducting gasis in direct contact with the lower substrate 104. The flexible seals106 and 108 move relative to the lower substrate 104 fin the normaldirection when the thermally conducting gas undergoes volumetric thermalexpansion. The upper substrate 102 deflects continuously and maximallyin the direction along the coolant fluid 110 flow direction when theflexible seals 106 and 108 deflect when the thermally conducting gasundergoes volumetric thermal expansion. The thermally conducting gas hashigh volumetric thermal expansion coefficient.

The flexible seals 106 and 108 allow a local expansion in the thin film100 heights due to both changes in internal pressure and the lower(heated) substrate 104 temperature. Similar effects are expected whenthe upper substrate 102 is a bimaterial plate separated from the lowersubstrate 104 via soft seals. The thin film 100 height varies linearlywith local pressure and local lower substrate 104 temperature accordingto the following relations:

$\begin{matrix}{{H(X)} = {\frac{h(x)}{h_{o}} = {1 + \frac{\overset{\_}{P}(X)}{S_{1}} + {F_{T\; 1}{\theta_{B}(X)}}}}} & {{Eq}.\mspace{11mu} 10}\end{matrix}$

where F_(T1) is the thermal expansion parameter, which is equal to:

$\begin{matrix}{F_{T\; 1} = {\beta \frac{q\; h_{o}}{k}}} & {{Eq}.\mspace{11mu} 11}\end{matrix}$

The coefficient β is thermal expansion coefficient of the flexiblecomplex seals 106 and 108. The parameter F_(T1) increases as the heatingload q, the thermal expansion coefficient β and the reference thin filmheight increase while it decreases as the fluid thermal conductivity kdecreases. The stiffness parameter s_(i) is generally related to theelastic properties of the flexible complex seals 106 and 108. When thethin film width is uniform (D=D₀), the velocity field (Equations 1, 2and 3) reduces to the following:

$\begin{matrix}{{{{U\left( {X,\eta,Z} \right)} = {\frac{u\left( {X,\eta,Z} \right)}{u_{o}} - {6\frac{\overset{\_}{P}}{X}{H^{2}\left( {\eta - \eta^{2}} \right)}}}};}{u_{o} = \frac{\left( {P_{i} - P_{e}} \right)h_{o}^{2}}{12\; \mu \; B}}} & {{Eq}.\mspace{11mu} 12} \\{{V\left( {X,\eta,Z} \right)} = {\left\lbrack {H(X)} \right\rbrack^{2}\left( \frac{\overset{\_}{P}}{X} \right)\left( \frac{H}{X} \right)\left( {\eta^{3} - \eta^{2}} \right)}} & {{Eq}.\mspace{11mu} 13} \\{{W\left( {X,\eta,Z} \right)} = 0} & {{Eq}.\mspace{11mu} 14}\end{matrix}$

where

${\xi = X},{\eta = {\frac{{Yh}_{o}}{h(x)}.}}$

As such, Equation 8 can be written as:

$\begin{matrix}{{{- 6}\frac{\overset{\_}{P}}{X}{H^{4}\left( {\eta - \eta^{2}} \right)}\frac{\partial\theta}{\partial\xi}} = {\frac{1}{{Pe}\; ɛ}\frac{\partial^{2}\theta}{\partial\eta^{2}}}} & {{Eq}.\mspace{11mu} 15}\end{matrix}$

where

$\frac{\overset{\_}{P}}{X}$

is evaluated from the following reduced form of the Reynolds equation:

$\begin{matrix}{{\frac{\partial}{\partial X}\left( {\left\lbrack {H(X)} \right\rbrack^{3}\frac{\partial\overset{\_}{P}}{\partial X}} \right)} = 0} & {{Eq}.\mspace{11mu} 16}\end{matrix}$

The uniform wall heat flux is assumed at the lower substrate while theupper substrate is considered to be insulated. This models the casewhere the fluidic thin film with a thin conductive lower substrate isplaced on the top of a heated surface while its upper substrate isconfigured from a less conductive flexible material (e.g. plastics). Theboundary conditions are

$\begin{matrix}{{{\theta \left( {X,0} \right)} = 0},{\left. \frac{\partial\theta}{\partial\eta} \right|_{X,{\eta = 0}} = {- {H(X)}}},{\left. \frac{\partial\theta}{\partial\eta} \right|_{X,{\eta = {11}}} = \left. {{- ɛ^{2}}{H(X)}\left( {{H(X)} - 1} \right)\frac{\partial\theta}{\partial\eta}} \middle| {}_{X,{\eta = {11}}}{\cong 0} \right.}} & {{Eq}.\mspace{11mu} 17}\end{matrix}$

The Nusselt number is defined as

$\begin{matrix}{{{{Nu}(X)} \equiv \frac{h_{c}h_{o}}{k}} = {\frac{1}{{\theta_{B}(X)} - {\theta_{m}(X)}} = \frac{1}{{\theta \left( {X,0} \right)} - {\theta_{m}(X)}}}} & {{Eq}.\mspace{11mu} 18}\end{matrix}$

where h_(c), θ_(m), and θ_(B) are the local convection coefficient,dimensionless mean bulk temperature and dimensionless lower substratetemperature, respectively, and are defined as follows:

$\begin{matrix}{{{\theta_{m}(X)} = {\frac{1}{{U_{m}(X)}H}{\int_{0}^{H}{{U\left( {X,Y} \right)}{\theta \left( {X,Y} \right)}{Y}}}}}{{U_{m}(X)} = {\frac{1}{H}{\int_{0}^{H}{{U\left( {X,Y} \right)}{Y}}}}}} & {{Eq}.\mspace{11mu} 19}\end{matrix}$

where U_(m)(X) is the average velocity inside the thin film at thedimensionless axial distance X.

Equations 15 and 16 are discretized using three points centraldifferencing in the transverse direction (η-direction) while two pointsdifferencing was utilized in the axial direction (X-direction). Thefinite difference equations for Equations 16 and 15 are the following,respectively:

$\begin{matrix}{\mspace{79mu} {{{\overset{\_}{P}}_{i - 1} - {\left\lbrack {1 + \left( \frac{H_{i + {1/2}}}{H_{i - {1/2}}} \right)^{3}} \right\rbrack {\overset{\_}{P}}_{i}} + {\left( \frac{H_{i + {1/2}}}{H_{i - {1/2}}} \right)^{3}{\overset{\_}{P}}_{i + 1}}} = 0}} & {{Eq}.\mspace{11mu} 20} \\{{{- 6}\left( \frac{{\overset{\_}{P}}_{i} - {\overset{\_}{P}}_{i - 1}}{\Delta\xi} \right){H_{i}^{4}\left( {\eta_{j} - \eta_{j}^{2}} \right)}\left( \frac{\theta_{i,j} - \theta_{{i - 1},j}}{\Delta \; \xi} \right)} = {\frac{1}{{Pe}\; ɛ}\left( \frac{\theta_{i,{j + 1}} - {2\theta_{i,j}} + \theta_{i,{j - 1}}}{\Delta \; \eta^{2}} \right)}} & {{Eq}.\mspace{11mu} 21}\end{matrix}$

where i and j are the location of the discretized point in the X and ηdirections respectively. The resulting tri-diagonal systems of algebraicequations, Equations 20 and 21 can be then solved utilizing, forexample, the well-established Thomas algorithm. The same procedure isrepeated for the consecutive i values of P _(i) and θ_(i,j) until jreaches the value M (M=101) at which X=1.0.

A thin film with h_(o)=200 μm and B=3 mm results in Peε=1.0 whenu_(o)=0.075 m/s (water as coolant) and it results in Peε=50 whenu_(o)=0.04 m/s (oil as coolant). It should be noted that a 100% increasein the heated substrate temperature relative to the inlet fluidtemperature, (T_(w)−T₁)/T₁=1.0, results in an expansion of the thin filmheight of the orders of 1.0<H(X=1.0)<2.0. The value of H(X=1.0)=2.0occurs when both lateral expansions and elastic forces in the flexiblecomplex seal are negligible as well as when an ideal gas is contained inthe closed cavities of these seals.

FIG. 2 illustrates a graph 200 depicting the variation of thedimensionless thin film height H with dimensionless axial distance X andthe thermal expansion parameter FT1. It is noticed that the thin filmheight increases as FT1 increases and that the maximum gradient of Hoccurs near the thin film inlet. This increases normal stresses due tobending in this region.

FIG. 3 illustrates a graph 300 depicting the variation of thedimensionless mean bulk temperature θ_(m) and dimensionless lowersubstrate temperature θ_(B) with the thermal expansion parameter FT1, inaccordance with the disclosed embodiments. The increase in FT1 (whenPeε=1.0 and S₁=5.0) enhances cooling inside the thin film where the meanbulk temperature θ_(m) at the exit is reduced by 70% by increasing FT1from FT1=0.0 to FT1=1.0. The lower substrate temperature θ_(B) at theexit is reduced by 25% when FT1 is changed from FT1=0.0 to FT1=1.0. Thereduction in θ_(B) is apparent when FT1<1.0 for the parameters shown inFIG. 3.

The expansion in the local thin film height and the lower substratetemperatures decrease as the Peclet number Pe or the aspect ratio εincrease as shown in the graphs 400 and 500 of FIGS. 4 and 5respectively. The decrease in H, θ_(m) and θ_(B) as Peε increases isapparent when Peε<10.

FIG. 6 illustrates a graph 600 depicting the variation of the lowersubstrate temperature at the exit θ_(B)(F_(T1),X=1) relative to thatwhen F_(T1)=0 with Peε and F_(T), in accordance with the disclosedembodiments. The flexible thin film can provide maximum enhancement inthe cooling effect of 45% and above (compared to the performanceordinary flexible fluidic thin films) as the heating load increases whenPeε is decreased below Peε=0.5. This indicates that thermally expandableflexible fluidic thin films are recommended to be used in Micro-ElectroMechanical Systems (MEMS) and electronic cooling applications. It isworth noting that increasing the thermal expansion parameter beyondcertain values will decrease the coolant velocity near the heatedsubstrate which can result in a reduction in the cooling enhancement ascan be seen from FIG. 6.

This fact can also be seen from FIG. 7, where the average Nusselt numberdecreases to a minimum and then it increases as FT1 increases. FIG. 7illustrates a graph 700 illustrating the, variation of the averageNusselt number Nu_(AVG)(F_(T1),X=1) relative to that when F_(T1)=0 withPeε and F_(T1), in accordance with the disclosed embodiments. For fulland stable utilization of thermally expandable flexible fluidic thinfilms, the thermal expansion parameter is recommended to be lower thanthe following critical value:

F _(T1)<(F _(T1))_(critical)=0.642Peε+2.363   Eq. 33

The following correlations are for (Nu)_(AVG) and (θ_(m))_(AVG) for thinfilms supported by flexible complex seals with flexible upper substratefor the specified range of parameters, 1.0<S₁<10, 1.0<Peε<50 and0<F_(T1)<1.0:

$\begin{matrix}{{{{Nu}_{AVG} = \frac{{0.0455\left( {{Pe}\; ɛ} \right)^{0.6793}F_{T\; 1}^{0.8345}} + {1.5285\left( {{Pe}\; ɛ} \right)^{0.2877}S_{1}^{0.3225}}}{\left( {{0.5169\mspace{14mu} S_{1}^{0.620}} + {1.255 \times 10^{- 3}F_{T\; 1}^{1.8691}}} \right)^{0.5503}}};}\mspace{20mu} {R^{2} = 0.996}} & {{Eq}.\mspace{11mu} 34} \\{{{\left( \theta_{m} \right)_{AVG} = \frac{{1.3709\left( {{Pe}\; ɛ} \right)^{- 0.7717}F_{T\; 1}^{1.6939}} + {1.1520\left( {{Pe}\; ɛ} \right)^{- 0.9898}S_{1}^{- 0.4380}}}{\left( {{2.1144\mspace{14mu} S_{1}^{- 0.3449}} + {1.8401\mspace{11mu} F_{T\; 1}^{1.3294}}} \right)^{2.4686}}};}\mspace{20mu} {R^{2} = 0.992}} & {{Eq}.\mspace{11mu} 35}\end{matrix}$

CONCLUSIONS

Heat transfer within flexible fluidic thin films that are expandable dueto pressure and heat can be analyzed as indicated herein. The uppersubstrate of the thin film can be assumed to be flexible and mobile. Theexpansion in the thin film heights is generally linearly related to thelocal fluid pressure and local lower substrate (e.g., heated). Thegoverning Reynolds, momentum and energy equations can be properlynon-dimensionalized and cast in proper forms. Such equations can be thensolved numerically utilizing an implicit computational method.

Parameters such as, for example, a stiffness parameter, a Peclet number,a thermal expansion parameter, an aspect ratio and the ratio of thewidth to the thin film length can be utilized as the main controllingparameters. The thin films produce significant increase in coolingcapacity as the heating load increases especially those operated withlower Peclet numbers. Finally, thermally expandable flexible fluidicthin films are recommended to be utilized in small sized thin films asfor Micro-Electro Mechanical Systems (MEMS) and electronic coolingapplications.

To the extent necessary to understand or complete the disclosure herein,all publications, patents, and patent applications mentioned herein areexpressly incorporated by reference therein to the same extent as thougheach are individually so incorporated.

Having thus described exemplary embodiments of the present invention, itshould be noted by those skilled in the art that the within disclosuresare exemplary only and that various other alternatives, adaptations, andmodifications may be made within the scope of the present invention.Accordingly, the present invention is not limited to the specificembodiments as illustrated herein, but is only limited by the followingclaims.

What is claimed is:
 1. An apparatus comprising: a first substrate and asecond substrate, said first substrate comprising an inflexiblesubstrate having a face in contact with at least one hot medium andhaving said other face in contact with a coolant fluid; said secondsubstrate comprising a flexible substrate having a face in contact withsaid coolant fluid and having said other face in contact with saidsurrounding ambient; the faces of the first and second substrates incontact with the coolant fluid are opposing each other; at least twocontinuous flexible seals attached to said faces of said first substrateand said second substrate opposing said coolant fluid to form at leastone closed enclosure comprising a thermally conducting gas; saidthermally conducting gas in direct contact with said first substrate;said flexible seal movable relative to said first substrate in saidnormal direction when said thermally conducting gas undergoes volumetricthermal expansion; a first at least one opening on said first substrateand a second at least one opening on said first substrate; said coolantfluid flowing between said first at least one opening and said second atleast one opening through said volume between said substrates excludingsaid volumes of said at least two continuous flexible seals and saidthermally conducting gas.
 2. The apparatus of claim 1, wherein saidsecond substrate deflects continuously and maximally in a directionalong a direction of said flow of said coolant fluid when said at leasttwo continuous flexible seals deflect when said thermally conducting gasundergoes volumetric thermal expansion.
 3. The apparatus as set forth inclaim 1, wherein said second substrate does not deflect or deflectsminimally in a direction normal to a direction of said flow of saidcoolant fluid when said at least two continuous flexible seals deflectand when said thermally conducting gas undergoes volumetric thermalexpansion.
 4. The apparatus as set forth in claim 1, wherein saidthermally conducting gas comprises a high volumetric thermal expansioncoefficient.